Monomial Bases for NBC Complexes Jason I. Brown Department of - - PDF document

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Monomial Bases for NBC Complexes

Jason I. Brown Department of Mathematics and Statistics Dalhousie University Halifax, NS B3H 3J5, CANADA Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027, USA sagan@math.msu.edu www.math.msu.edu/˜sagan

• 1. Complexes and chromatic polynomials
• 2. NBC complexes
• 3. Stanley-Reisner rings and hsop’s
• 4. Monomial ideals
• 5. Comments

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• 1. Complexes and chromatic polynomials

Let ∆ be a simplicial complex on a finite set E, so ∆ is a family of subsets of E satisfying S ∈ ∆ and T ⊆ S implies T ∈ ∆. The S ∈ ∆ are called faces. We assume ∆ is pure

• f rank r meaning that |S| = r for all maximal faces

S ∈ ∆. For 0 ≤ i ≤ r, let fi = fi(∆) = # of faces S ∈ ∆ with |S| = i. The f-polynomial of ∆ is f(x) = f0 + f1x + f2x2 + · · · + frxr. The h-polynomial of ∆ is h(x) = (1 − x)rf

• x

1 − x

• =

f0(1 − x)r + f1x(1 − x)r−1 + · · · + frxr. and let hi = hi(∆) = coefficient of xi in h(x).

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Let G = (V, E) be a graph with |V | = p and |E| = q. A proper coloring of G is c : V → {1, 2, . . . , λ} such that vw ∈ E implies c(v) = c(w). The chromatic polynomial of G is P(G) = P(G; λ) = # of such proper colorings.

• Example. Let

G =

④

t

④

w

④u ④

v

❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅

P(G) = # ways to color t, then u, then v, then w = λ(λ − 1)(λ − 2)(λ − 2) = λ4 − 5λ3 + 8λ2 − 4λ. Proposition 1 Let Kp be the edgeless graph and let T be a tree on p vertices. Then P(Kp; λ) = λp and P(T; λ) = λ(λ − 1)p−1.

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Let G be a graph and e ∈ E. Let G\e = G with e deleted, G/e = G with e contracted.

• Example. Let

G =

① ①

w

① ①

v

❅ ❅ ❅

• e

❅ ❅ ❅

G\e =

① ①

w

① ①

v

❅ ❅ ❅

• ❅

❅ ❅

G/e =

① ① ①

v = w

• ❅

❅ ❅

Theorem 2 (Deletion-Contraction) For e ∈ E P(G) = P(G\e) − P(G/e)

• Proof. If e = vw then

P(G\e) = (# proper c for G\e s.t. c(v) = c(w)) +(# proper c for G\e s.t. c(v) = c(w)) = P(G) + P(G/e). Corollary 3 For any graph G:

• 1. P(G; λ) is a monic polynomial in λ.
• 2. deg P(G; λ) = p = |V |.
• 3. Coefficients of P(G; λ) alternate in sign.

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• 2. NBC complexes

Define coefficients fi by P(G; λ) = f0λp − f1λp−1 + · · · and coefficients hi by P(G; λ) = h0λ(λ − 1)p−1 − h1λ(λ − 1)p−2 + · · · . Let C = C(G) = set of cycles/circuits of G. Let G be ordered meaning that E has been given a total order e1 < e2 < . . . < eq. Then each C ∈ C has broken circuit C = C − min C. The NBC complex of G is ∆ = ∆(G) = {S ⊆ E : S contains no C}. Then ∆(G) is a pure simplicial complex. Theorem 4 Let P(G; λ) have coefficients fi and hi as defined above. Then for 0 ≤ i ≤ p fi = fi(∆(G)) and hi = hi(∆(G)).

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G =

④ ④ ④ ④ ❅ ❅ ❅ ❅

5

• 1
• 2

3

❅ ❅ ❅ ❅ 4

C(G) =

④ ④ ④ ❅ ❅ ❅ ❅

5

• 2

3

④ ④ ④

• 1

3

❅ ❅ ❅ ❅ 4 ④ ④ ④ ④ ❅ ❅ ❅ ❅

5

• 1
• 2

❅ ❅ ❅ ❅ 4

¯ C(G) = {35, 34, 245} ∆(G) = {∅} ∪ {1, 2, 3, 4, 5} ∪{12, 13, 14, 15, 23, 24, 25, 45} ∪{123, 124, 125, 145} (fi(∆)) = (1, 5, 8, 4, 0). P(G; λ) = λ(λ − 1)(λ − 2)2 = λ4 − 5λ3 + 8λ2 − 4λ. P

• t

t t t ❅ ❅

• ❅

❅ ; λ

• = P
• t

t t t ❅ ❅

• ❅

❅ ; λ

• − P
• t

t t

• ❅

❅ ; λ

• = P
• t

t t t ❅ ❅ ; λ

• − P
• t

t t ❅ ❅ ; λ

• − P
• t

t t

• ; λ
• + P
• t

t

; λ

• = λ(λ − 1)3 − λ(λ − 1)2 − λ(λ − 1)2 + λ(λ − 1)

= λ(λ − 1)3 − 2λ(λ − 1)2 + λ(λ − 1) (hi(∆)) = (1, 2, 1, 0, 0)

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• 3. Stanley-Reisner rings and hsop’s

Let F[x] be the polynomial ring over field F with variables x = {x1, . . . , xq}. If E = {e1, . . . , eq} then S ⊆ E has monomial

xS =

• ei∈S

xi. Simplicial complex ∆ has Stanley-Reisner ring F(∆) = F[x]/(xS : S ∈ ∆). In particular, for an ordered graph G we let F(G) = F(∆(G)) = F[x]/(xC : C ∈ C(G)). Now F(G) has a homogeneous system of parame- ters (hsop) of degree one θ1, . . . , θt, i.e.,

• 1. θi is linear without constant term for all i,
• 2. θ1, . . . , θt are algebraically independent,
• 3. F(G)/(θ1, . . . , θt) is finite dim. over F.

Brown gave an explicit hsop for F(G). WLOG G is connected and let T be a spanning tree of G. If e ∈ E(T) then e has fundamental disconnecting set De = De(G) = {f ∈ E(G) : T − e + f connected} and hsop element (when F = Z2) θe =

• ei∈De

xi.

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Example G =

✉ ✉ ✉ ✉ ✉ ✉

1

• 2

3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

T =

✉ ✉ ✉ ✉ ✉ ✉

3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

C(G) = {13467, 2345, 12567}

Z2(G)

= Z2[x1, . . . , x7]/(x3x4x6x7, x3x4x5, x2x5x6x7)

✉ ✉ ✉ ✉ ✉ ✉

1

• 2

3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

θ3 = x3 + x1 + x2

✉ ✉ ✉ ✉ ✉ ✉

1

• 2

3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

θ4 = x4 + x1 + x2

✉ ✉ ✉ ✉ ✉ ✉

2 3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

θ5 = x5 + x2

✉ ✉ ✉ ✉ ✉ ✉

1

• 3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

θ6 = x6 + x1

✉ ✉ ✉ ✉ ✉ ✉

1

• 3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

θ7 = x7 + x1

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• 4. Monomial ideals

If F(∆) has an hsop θ1, . . . , θt we let R(∆) = F(∆)/(θ1, . . . , θt). Consider Mon(k) = set of monomials in F[x1, . . . , xk]. A subset L ⊆ Mon(k) is a lower order ideal if m ∈ L and n|m imples n ∈ L. The lower order ideal generated by S ⊆ Mon(k) is L(S) = {m ∈ Mon(k) : m|n for some n ∈ S}. Upper ideal and U(S) are defined dually. Theorem 5 (Macaulay, Stanley) Suppose ∆ is a simplicial complex and that the ring F(∆) is Cohen-

• Macaulay. Then R(∆) has a basis, L, which is a

lower order ideal of monomials and hi(∆) = # of monomials of total degree i in L.

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For a graph G, F(G) is Cohen-Macaulay. We have a conjectured construction of a basis for R(G). An ordering e1 < . . . < eq is standard if the last p−1 edges form a tree. Let k = |E(G) − E(T)|. We can pick the monomial basis for R(G) inside Mon(k) since Brown’s θi can be used to eliminate the other variables, replacing each xC by a polynomial pC.

• Example. In our running example, k = 2 and

Z2(G) = Z2[x1, . . . , x7]/(x3x4x6x7, x3x4x5, x2x5x6x7).

θ3 = x3 + x1 + x2, θ4 = x4 + x1 + x2, θ5 = x5 + x2. So, picking one of the broken circuit monomials

xC = x3x4x5

becomes pC = (x1 + x2)2x2. For 1 ≤ i ≤ k, the graph T + ei has a unique fun- damental circuit Ci. Conjecture 6 Let G be connected. Then there is a standard ordering of E such that R(G) has basis L(G) = Mon(k) − U(mC : C ∈ C(G)) where mC =

  

x#Ci

i

if C = Ci fundamental, min pC else. Here min p picks out the lexicographically smallest monomial in p.

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G =

① ① ① ① ① ①

1

• 2

3

◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑✑✑ ✑

5 6

❅ ❅ ❅ ❅ ❅ ❅

7 T =

① ① ① ① ① ①

3

◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑✑✑ ✑

5 6

❅ ❅ ❅ ❅ ❅ ❅

7 Fundamental cycles:

✉ ✉ ✉ ✉ ✉ ✉

1

• 3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

C1 = {1, 3, 4, 6, 7} mC1 = x4

1

✉ ✉ ✉ ✉ ✉ ✉

2 3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

C2 = {2, 3, 4, 5} mC2 = x3

2

Nonfundamental cycle:

✉ ✉ ✉ ✉ ✉ ✉

1

• 2

3

◗ ◗ ◗ ◗ ◗ ◗

4

✑✑✑✑✑✑

5 6

❅ ❅ ❅ ❅

7

C3 = {1, 2, 5, 6, 7}

xC3

= x2x5x6x7 pC3 = x2x2x1x1 mC3 = x2

1x2 2

So R(G) has basis L(G) = Mon(2) − U(x4

1, x3 2, x2 1x2 2).

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• 5. Comments

A graph with a standard ordering satisfying the conjecture is said to have a broken circuit basis.

• a. (Generalized) theta graphs and phi graphs have

broken circuit bases.

• b. By only considering the fundamental circuits:

Proposition 7 If G has a broken circuit basis and ci = |Ci| for 1 ≤ i ≤ k, then R(G) is spanned by L

 

• 1≤i≤k

xci−2

i

  .

Stanley showed that the number of acyclic orien- tations of G is given by P(G; −1). So one can use this proposition to estimate their number.

• c. The results we have about broken circuit bases

are proved by deletion/contraction. It is hoped that together with the ear decomposition of a block we will be able to prove the full conjecture. d. The conjecture may even be true for repre- sentable matroids.

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